SLAC - PUB - 3788 September 1985 CT) THE ENSEMBLE PROJECTOR MONTE CARLO METHOD, STUDYING THE LATTICE SCHWINGER MODEL IN THE HAMILTONIAN FORMULATION* J. RANFT+ Stanford Linear Accelerator Center Stanford University, Stanford, California, 94905 and A. S CHILLER Sektion Physik, Karl-Marx- Universita”t Leipzig, G.D.R. Submitted to Physical Review D * Work supported by the Department of Energy, contract DE - AC03 - 76SF00515. + Permanent Address: Sektion Physik, Karl-Marx-Universitgt, Leipzig, G.D.R.
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SLAC - PUB - 3788 September 1985 CT)
THE ENSEMBLE PROJECTOR MONTE CARLO METHOD, STUDYING THE LATTICE SCHWINGER MODEL IN
THE HAMILTONIAN FORMULATION*
J. RANFT+
Stanford Linear Accelerator Center
Stanford University, Stanford, California, 94905
and
A. S CHILLER
Sektion Physik, Karl-Marx- Universita”t
Leipzig, G.D.R.
Submitted to Physical Review D
* Work supported by the Department of Energy, contract DE - AC03 - 76SF00515. + Permanent Address: Sektion Physik, Karl-Marx-Universitgt, Leipzig, G.D.R.
ABSTRACT
The ensemble projector Monte Carlo method is a promising method to study
lattice gauge theories with fermions in the Hamiltonian formulation. We study
the massive Schwinger model and show, that consistent results are obtained in
the presence of positive and negative matrix elements. The expectation values for
the average energy calculated from matrix elements with negative and positive
scores, and calculated from the average scores are consistent with each other
and with results obtained from the local Hamiltonian Monte Carlo method. In
contrast to the latter method, the ensemble projector Monte Carlo method can
be applied also to gauge field theories in 2+1 and 3+1 dimensions.
1. INTRODUCTION
The introduction of more effective methods for the Monte Carlo simulation of
gauge field theories with fermions is an important problem in present day lattice
gauge field theories. Most of the Monte Carlo algorithms used at present start
from the Wilson formulation of lattice gauge theories.’ A second method is the
Hamiltonian formulation of lattice gauge theories,” where the time remains a
continuous variable and the formulation is in terms of the lattice Hamiltonian
and lattice eigenstates.
The local Hamiltonian Monte Carlo method3 is an effective Hamiltonian
method to treat lattice theories with fermions in d = 1 + 1 dimensions. This
method was applied to model gauge field theories, like the massless and massive5
Schwinger model, models with gauge bosons, Higgs bosons and fermions6 and
supersymmetric models.’ Unfortunately, it is not possible to extend this method
to models with more than one spatial dimension.
The projector Monte Carlo method is a new Hamiltonian Monte Carlo method8
which was applied to lattice models8 and to pure lattice gauge theories in
d = 2 + 1 dimensions,g to the Schwinger model with fermionsl’ and to the
.d = 1 + 1 SU(2) lattice gauge theory. l1 This method is however rather ineffective
especially for large lattices due to the large fluctuations of the scores which go into
the calculated expectation values. Because of this, this method was improved in
several ways: 8,12-14 The parallel scores method was shown to give good results
for the d = 2 + 1 U(1) gauge theory without fermions. Another generalization is
the ensemble projector method,13’14 which was applied to the pure U(1) lattice
gauge theory in 2+115 and 3+113 dimensions, to a study of the string tension
and of screening in the Schwinger model16 and to the d = 1 + 1 SU(2)-lattice
3
gauge theory with fermions. l7 Here we continue to study and use this method
for the Schwinger model in d = 1 + 1 dimensions. the aim of the present paper
is, to study the calculation of expectation values with this method in a situation,
where matrix elements and scores can have negative signs. This happens in the
Schwinger model, if we does not restrict the calculation to lattice configurations
periodic in time direction as was done using the local Hamiltonian Monte Carlo
method. 3-5
It was also shown in Ref. 18, that the presence of intrinsic negative signs in
the matrix elements does not prevent the Hamiltonian Monte Carlo calculation.
The same problem will occur again in Hamiltonian lattice gauge theories with
fermions in d = 2 + 1 and d = 3 + 1 dimensions, which we did start to study.”
In Section 2 we present the ensemble projector Monte Carlo method as applied
in this paper, in Section 3 we apply the method to the massive Schwinger model.
In Section 4 we present and discuss the results.
2. THE ENSEMBLE PROJECTOR MONTE CARLO METHOD
We introduce a lattice Hamiltonian H defined on a discrete spatial lattice,
14, I+>, etc. are lattice eigenstates. The operator exp(-/3H) is a projection
operator to the lowest energy eigenstate with given quantum numbers.
Provided that Ix) and 14) are states not orthogonal to the lattice ground
state I$) we calculate expectation values of operators as follows:
(1)
4
or
olv2l+> = 25% (xlemPH GWPHI& (xle-2PH14) * (2)
To calculate the matrix elements in (1) and (2) one splits /3 in L intercalls Ar
p=LAr (3)
and furthermore the Hamiltonian into two (one more) parts
H = H1 i-H2 . (4)
This splitting is arbitrary provided it leads to local matrix elements down in
expression (6) h’ h w IC contain only the variables of one lattice point and its nearest
neighbors. We define Uk = exp(-ArHk) and find
e -ATH=UlU2 l-f AT~[H~,H~]+... >
(5)
-.that for sufficiently small Ar the terms in Ar2 can be neglected. We obtain for